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namespace OAI
namespace QuarticPowerFree
section
open MvPolynomial
open scoped BigOperators Classical
theorem quotient_pair_generates {n d : ℕ} (a b : Fin d → MvPolynomial (Fin n) ℂ)
(hg : Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤)
(I : Ideal (MvPolynomial (Fin n) ℂ)) :
Algebra.adjoin ℂ (Set.range (fun i => Ideal.Quotient.mk I (a i)) ∪
Set.range (fun i => Ideal.Quotient.mk I (b i))) = ⊤ := by
let e := Ideal.Quotient.mkₐ ℂ I
have hr : Set.range (fun i => Ideal.Quotient.mk I (a i)) ∪ Set.range (fun i => Ideal.Quotient.mk I (b i)) =
e '' (Set.range a ∪ Set.range b) := by
simp only [Set.image_union,← Set.range_comp]
rfl
rw [hr,Algebra.adjoin_image,hg]
apply top_unique
intro z _
obtain ⟨p,rfl⟩ := Ideal.Quotient.mkₐ_surjective ℂ I z
exact ⟨p,by trivial,_root_.rfl⟩
theorem norm_dimension_le {n d : ℕ} (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime]
(k : ℕ) (a b : Fin d → MvPolynomial (Fin n) ℂ) (q : MvPolynomial (Fin n) ℂ) (θ : Fin d → ℂ)
(ha : ∀ i, Ideal.Quotient.mk I (a i) ≠ 0)
(heq : ∀ i, Ideal.Quotient.mk I (a i)^k*Ideal.Quotient.mk I (b i) =
Ideal.Quotient.mk I q-algebraMap ℂ _ (θ i))
(hg : Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤)
(hq : Transcendental ℂ (Ideal.Quotient.mk I q)) : affineDimension I ≤ d+1 := by
let := pair_projection_isAlgebraic k (fun i => Ideal.Quotient.mk I (a i))
(fun i => Ideal.Quotient.mk I (b i)) (Ideal.Quotient.mk I q) θ ha heq (quotient_pair_generates a b hg I)
obtain ⟨m,j,_,hj,ht⟩ := left_coordinates_transcendence_selection
(fun i => Ideal.Quotient.mk I (a i)) (Ideal.Quotient.mk I q) hq
have hm : m ≤ d := by simpa using Fintype.card_le_of_injective j hj
have hdim := hilbert_natDegree_eq_trdeg I (coordinateHilbert I) (coordinateHilbert_eventually I)
rw [ht] at hdim
have hh : affineDimension I = m+1 := by exact_mod_cast hdim
omega
theorem polynomial_constant_at_point {n : ℕ} (I : Ideal (MvPolynomial (Fin n) ℂ))
(q : MvPolynomial (Fin n) ℂ) (t : ℂ)
(ht : algebraMap ℂ (MvPolynomial (Fin n) ℂ ⧸ I) t = Ideal.Quotient.mk I q)
(z : Fin n → ℤ) (hz : PointOn I z) : MvPolynomial.aeval (fun j => (z j:ℂ)) q = t := by
have hh := congrArg (pointEvaluation I z hz) ht
rw [AlgHom.commutes] at hh
change t = MvPolynomial.aeval (fun j => (z j:ℂ)) q at hh
exact hh.symm
theorem algebraic_polynomial_fibre_count {n : ℕ} (S : Finset (Fin n → ℤ)) (F : ℕ)
(q : MvPolynomial (Fin n) ℂ)
(hf : ∀ t : ℂ, (S.filter (fun z => MvPolynomial.aeval (fun j => (z j:ℂ)) q = t)).card ≤ F)
(I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime]
(hq : IsAlgebraic ℂ (Ideal.Quotient.mk I q)) : (S.filter (PointOn I)).card ≤ F := by
obtain ⟨t,ht⟩ := algebraic_coordinate_constant (Ideal.Quotient.mk I q) hq
apply (Finset.card_le_card ?_).trans (hf t)
intro z hz
have h := Finset.mem_filter.mp hz
exact Finset.mem_filter.mpr ⟨h.1,polynomial_constant_at_point I q t ht z h.2⟩
theorem algebraic_left_fibre_count {n d : ℕ} (S : Finset (Fin n → ℤ)) (F : ℕ)
(x : Fin d → Fin n)
(hf : ∀ c : Fin d → ℤ, (S.filter (fun z => ∀ i, z (x i) = c i)).card ≤ F)
(I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime]
(hx : ∀ i, IsAlgebraic ℂ (Ideal.Quotient.mk I (MvPolynomial.X (x i)))) :
(S.filter (PointOn I)).card ≤ F := by
by_cases he : (S.filter (PointOn I)).Nonempty
· obtain ⟨z,hz⟩ := he
have hzI := (Finset.mem_filter.mp hz).2
apply (Finset.card_le_card ?_).trans (hf (fun i => z (x i)))
intro w hw
obtain ⟨hwS,hwI⟩ := Finset.mem_filter.mp hw
refine Finset.mem_filter.mpr ⟨hwS,fun i => ?_⟩
obtain ⟨t,ht⟩ := algebraic_coordinate_constant (Ideal.Quotient.mk I (MvPolynomial.X (x i))) (hx i)
exact Int.cast_injective (α := ℂ) ((coordinate_constant_at_point I w hwI (x i) t ht).trans
(coordinate_constant_at_point I z hzI (x i) t ht).symm)
· rw [Finset.not_nonempty_iff_eq_empty.mp he,Finset.card_empty]
exact Nat.zero_le _
end
section
open MvPolynomial
theorem norm_generic_threshold_uniform {n d h : ℕ} (hd : 5 ≤ d) (hd' : d ≤ 8)
(hh : 2 ≤ h) (hh' : h ≤ d) (E D : ℕ) (hD : 0 < D) :
∃ C : ℝ, 0 ≤ C ∧
∀ (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime]
(a b : Fin d → MvPolynomial (Fin n) ℂ) (q : MvPolynomial (Fin n) ℂ)
(θ : Fin d → ℂ), Function.Injective θ →
(∀ i, Ideal.Quotient.mk I (a i) ≠ 0) →
(∀ i, Ideal.Quotient.mk I (a i)^(d-2)*Ideal.Quotient.mk I (b i) =
Ideal.Quotient.mk I q-algebraMap ℂ _ (θ i)) →
Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤ →
Transcendental ℂ (Ideal.Quotient.mk I q) → affineDimension I = h → affineDegree I ≤ E →
(∀ i, (a i).totalDegree ≤ D) → q.totalDegree ≤ D →
∀ (w : Fin n → ℕ), (∀ i, w i ≠ 0) →
(∀ i, a i ∈ weightedSpace w 5) → (∀ i, b i ∈ weightedSpace w 6) →
∀ (ω : Fin n → ℝ), (∀ i, 0 < ω i) → (∀ i, (w i:ℝ) = 15*ω i) →
∀ T : ℝ, 1 ≤ T →
T^h/((h.factorial:ℝ)*(normBoxEnvelope d h)^h)-C*T^(h-1) ≤ weightedHilbert I ω T := by
have hsucc : h-1+1 = h := by omega
obtain ⟨C,hC,hbound⟩ := norm_real_hilbert_uniform (n := n) (d := d) (h := h-1) (K := ℂ)
(k := d-2) (by omega) 5 6 (by norm_num) (by norm_num) E D hD 15 (by norm_num)
refine ⟨C,hC,fun I _ a b q θ hθ ha heq hg hq hdim hdeg hadeg hqdeg w hw haw hbw ω hω hscale T hT => ?_⟩
have hb := hbound I a b q θ hθ ha heq (quotient_pair_generates a b hg I) hq
(hsucc ▸ hdim) (hsucc ▸ hh') hdeg hadeg hqdeg w hw haw hbw ω hω hscale T hT
rw [hsucc] at hb
norm_num only [Nat.min_def,show (5:ℕ) ≤ 6 by omega,ite_true,Nat.cast_ofNat] at hb
have hc := mul_le_mul_of_nonneg_right (normBox_generic_threshold d h hd hd' hh hh')
(pow_nonneg (by linarith : 0 ≤ T) h)
have hc' : T^h/((h.factorial:ℝ)*(normBoxEnvelope d h)^h) ≤
5*(((d-2)*5+6:ℕ):ℝ)^(h-1)*15^h/((h.factorial:ℝ)*5^h*6^h)*T^h := by
simpa only [one_div_mul_eq_div] using hc
linarith only [hb,hc']
theorem norm_full_threshold_uniform {n d : ℕ} (hd : 5 ≤ d) (hd' : d ≤ 8) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime]
(a b : Fin d → MvPolynomial (Fin n) ℂ) (q : MvPolynomial (Fin n) ℂ)
(θ : Fin d → ℂ),
(∀ i, Ideal.Quotient.mk I (a i) ≠ 0) →
(∀ i, Ideal.Quotient.mk I (a i)^(d-2)*Ideal.Quotient.mk I (b i) =
Ideal.Quotient.mk I q-algebraMap ℂ _ (θ i)) →
Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤ →
Transcendental ℂ (Ideal.Quotient.mk I q) → affineDimension I = d+1 →
∀ (w : Fin n → ℕ), (∀ i, w i ≠ 0) →
(∀ i, a i ∈ weightedSpace w 5) → (∀ i, b i ∈ weightedSpace w 6) →
∀ (ω : Fin n → ℝ), (∀ i, 0 < ω i) → (∀ i, (w i:ℝ) = 15*ω i) →
∀ T : ℝ, 1 ≤ T →
T^(d+1)/(((d+1).factorial:ℝ)*(normBoxEnvelope d (d+1))^(d+1))-C*T^d ≤ weightedHilbert I ω T := by
have hsucc : d-1+1 = d := by omega
have hsucc' : d-1+2 = d+1 := by omega
have hb := norm_full_real_hilbert_uniform (N := n) (n := d-1) (K := ℂ)
(d-2) 5 6 (by norm_num) (by norm_num) 15 (by norm_num)
rw [hsucc,hsucc'] at hb
obtain ⟨C,hC,hbound⟩ := hb
refine ⟨C,hC,fun I _ a b q θ ha heq hg hq hdim w hw haw hbw ω hω hscale T hT => ?_⟩
have hb := hbound I a b q θ ha heq (quotient_pair_generates a b hg I) hq
hdim w hw haw hbw ω hω hscale T hT
have hc := mul_le_mul_of_nonneg_right (normBox_full_threshold d hd hd')
(pow_nonneg (by linarith : 0 ≤ T) (d+1))
have hc' : T^(d+1)/(((d+1).factorial:ℝ)*(normBoxEnvelope d (d+1))^(d+1)) ≤
(((d-2)*5+6:ℕ):ℝ)^(d-1)*15^(d+1)/(((d+1).factorial:ℝ)*5^d*6^d)*T^(d+1) := by
simpa only [one_div_mul_eq_div] using hc
norm_num only [Nat.cast_ofNat] at hb
linarith only [hb,hc']
open MvPolynomial
open scoped BigOperators Classical
structure NormBoxSystem (n d D : ℕ) (ω : Fin n → ℝ) (w : Fin n → ℕ) where
Carrier : Type
[field : Field Carrier]
[charZero : CharZero Carrier]
v : Fin d → Carrier
independent : LinearIndependent ℚ v
σ : Fin d → Carrier →+* ℂ
θ : Fin d → ℂ
distinct : Function.Injective θ
x : Fin d → Fin n
xweight : ∀ i, ω (x i) = 1/3
b : Fin d → MvPolynomial (Fin n) ℂ
q : MvPolynomial (Fin n) ℂ
generates : Algebra.adjoin ℂ
(Set.range (fun i => coordinateLinearForm x (σ i ∘ v)) ∪ Set.range b) = ⊤
leftDegree : ∀ i, (coordinateLinearForm x (σ i ∘ v)).totalDegree ≤ D
qDegree : q.totalDegree ≤ D
leftWeight : ∀ i, coordinateLinearForm x (σ i ∘ v) ∈ weightedSpace w 5
rightWeight : ∀ i, b i ∈ weightedSpace w 6
attribute [instance] NormBoxSystem.field NormBoxSystem.charZero
noncomputable def NormBoxSystem.a {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ}
(M : NormBoxSystem n d D ω w) (i : Fin d) : MvPolynomial (Fin n) ℂ :=
coordinateLinearForm M.x (M.σ i ∘ M.v)
noncomputable def NormBoxSystem.equation {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ}
(M : NormBoxSystem n d D ω w) (i : Fin d) : MvPolynomial (Fin n) ℂ :=
M.a i^(d-2)*M.b i-M.q+C (M.θ i)
noncomputable def NormBoxSystem.ideal {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ}
(M : NormBoxSystem n d D ω w) : Ideal (MvPolynomial (Fin n) ℂ) :=
Ideal.span (Set.range M.equation)
theorem NormBoxSystem.quotient_equations {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ}
(M : NormBoxSystem n d D ω w) (I : Ideal (MvPolynomial (Fin n) ℂ)) (hI : M.ideal ≤ I) :
∀ i, Ideal.Quotient.mk I (M.a i)^(d-2)*Ideal.Quotient.mk I (M.b i) =
Ideal.Quotient.mk I M.q-algebraMap ℂ _ (M.θ i) := by
intro i
have hm : M.equation i ∈ I := hI (Ideal.subset_span ⟨i,rfl⟩)
have he := (Ideal.Quotient.eq_zero_iff_mem).mpr hm
simp only [NormBoxSystem.equation,map_add,map_sub,map_mul,map_pow] at he
change Ideal.Quotient.mk I (M.a i)^(d-2)*Ideal.Quotient.mk I (M.b i)-
Ideal.Quotient.mk I M.q+algebraMap ℂ (MvPolynomial (Fin n) ℂ ⧸ I) (M.θ i) = 0 at he
linear_combination he
theorem NormBoxSystem.quotient_left_nonzero {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ}
(M : NormBoxSystem n d D ω w) (I : Ideal (MvPolynomial (Fin n) ℂ))
(z : Fin n → ℤ) (hz : PointOn I z)
(ha : ∀ i, MvPolynomial.aeval (fun j => (z j:ℂ)) (M.a i) ≠ 0) :
∀ i, Ideal.Quotient.mk I (M.a i) ≠ 0 := by
intro i hi
have hh := congrArg (pointEvaluation I z hz) hi
rw [map_zero] at hh
change MvPolynomial.aeval (fun j => (z j:ℂ)) (M.a i) = 0 at hh
exact ha i hh
theorem NormBoxSystem.right_real_weight {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ}
(M : NormBoxSystem n d D ω w) (hω : ∀ i, 0 < ω i) (hscale : ∀ i, (w i:ℝ) = 15*ω i)
(I : Ideal (MvPolynomial (Fin n) ℂ)) (i : Fin d) :
Ideal.Quotient.mk I (M.b i) ∈ weightedFunctions I ω (2/5) := by
have hh := weightedSpace_map_le_weightedFunctions I w ω hω 15 (by norm_num) hscale (2/5) (by norm_num)
norm_num only [show (15:ℝ)*(2/5) = 6 by norm_num,Nat.floor_natCast] at hh
exact hh ⟨M.b i,M.rightWeight i,rfl⟩
end
open MvPolynomial
open scoped BigOperators Classical
theorem norm_component_alternatives {n d h D : ℕ} (hd : 5 ≤ d) (hd' : d ≤ 8) (hh : 1 ≤ h) (hh' : h ≤ d+1) (hD : 0 < D)
(ω : Fin n → ℝ) (hω : ∀ i, 0 < ω i) (hωW : ∀ i, ω i ≤ 2/5)
(w : Fin n → ℕ) (hw : ∀ i, w i ≠ 0) (hscale : ∀ i, (w i:ℝ) = 15*ω i)
(M : NormBoxSystem n d D ω w) (Cbox : ℝ) (hCbox : 0 < Cbox)
(F E : ℕ) (ε : ℝ) (hε : 0 < ε) :
∃ K H : ℝ, 1 ≤ K ∧ 0 ≤ H ∧ ∀ X : ℝ, 2 ≤ X → ∀ S : Finset (Fin n → ℤ),
(∀ z ∈ S, ∀ j, |(z j:ℝ)| ≤ Cbox*X^(ω j)) →
(∀ t : ℂ, (S.filter (fun z => MvPolynomial.aeval (fun j => (z j:ℂ)) M.q = t)).card ≤ F) →
(∀ c : Fin d → ℤ, (S.filter (fun z => ∀ i, z (M.x i) = c i)).card ≤ F) →
∀ (I : Ideal (MvPolynomial (Fin n) ℂ)), I.IsPrime → M.ideal ≤ I →
affineDimension I = h → affineDegree I ≤ E →
(∀ i, Ideal.Quotient.mk I (M.a i) ≠ 0) →
((S.filter (PointOn I)).card:ℝ) ≤ K*X^ε ∨
∀ T : ℝ, 1 ≤ T → T^h/((h.factorial:ℝ)*(normBoxEnvelope d h)^h)-H*T^(h-1) ≤ weightedHilbert I ω T := by
classical
have hsmall (X : ℝ) (hX : 2 ≤ X) (S : Finset (Fin n → ℤ)) (I : Ideal (MvPolynomial (Fin n) ℂ))
(K : ℝ) (hKF : (F:ℝ) ≤ K) (hK : 0 ≤ K) (hcard : (S.filter (PointOn I)).card ≤ F) :
((S.filter (PointOn I)).card:ℝ) ≤ K*X^ε := by
have hx : 1 ≤ X^ε := Real.one_le_rpow (by linarith) hε.le
calc
_ ≤ (F:ℝ) := by exact_mod_cast hcard
_ ≤ K := hKF
_ ≤ K*X^ε := by nlinarith
by_cases h1 : h = 1
· subst h
obtain ⟨B,H,hB,hH,hcurve⟩ := norm_curve_alternatives (n := n) (d := d) (by omega)
(d-2) (by omega) ω hω M.x (1/3) (2/5) (2/5) Cbox (by norm_num) (by norm_num)
(by norm_num) hCbox M.xweight hωW E ε hε
let K : ℝ := max 1 (max B F)
have hK : 1 ≤ K := le_max_left _ _
have hKF : (F:ℝ) ≤ K := (le_max_right _ _).trans (le_max_right _ _)
have hKB : B ≤ K := (le_max_left _ _).trans (le_max_right _ _)
refine ⟨K,H,hK,hH,fun X hX S hbox hnF haF I hI hMI hdim hdeg ha => ?_⟩
let : I.IsPrime := hI
by_cases hq : IsAlgebraic ℂ (Ideal.Quotient.mk I M.q)
· exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_polynomial_fibre_count S F M.q hnF I hq))
by_cases hxa : ∀ i, IsAlgebraic ℂ (Ideal.Quotient.mk I (MvPolynomial.X (M.x i)))
· exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_left_fibre_count S F M.x haF I hxa))
push Not at hxa
obtain ⟨s,hs⟩ := hxa
have hsS (z) (hz : z ∈ S.filter (PointOn I)) := (Finset.mem_filter.mp hz).1
have hh := hcurve M.Carrier M.v M.independent M.σ M.θ M.distinct X hX
(S.filter (PointOn I)) (fun z hz j => hbox z (hsS z hz) j) I hI hdim.le hdeg
(fun z hz => (Finset.mem_filter.mp hz).2) M.b M.q
(M.right_real_weight hω hscale I) hq (M.quotient_equations I hMI) s hs
rcases hh with hc | hHilb
· left
exact hc.trans (mul_le_mul_of_nonneg_right hKB (Real.rpow_nonneg (by linarith) _))
· right
intro T hT
have hb := hHilb T (by linarith)
have hc := mul_le_mul_of_nonneg_right (normBox_curve_threshold d hd hd') (by linarith : 0 ≤ T) norm_num only [pow_one,Nat.factorial_one,Nat.cast_one,one_mul,Nat.sub_self,pow_zero,mul_one]
have hc' : T/normBoxEnvelope d 1 ≤ min (2/(1/3:ℝ)) (max (1/(1/3:ℝ)) (((d-1)*(d-2-1):ℕ)/(2/5:ℝ)))*T := by
simpa only [one_div_mul_eq_div] using hc
linarith only [hb,hc']
· have hh2 : 2 ≤ h := by omega
by_cases hhd : h ≤ d
· obtain ⟨H,hH,hHilb⟩ := norm_generic_threshold_uniform (n := n) hd hd' hh2 hhd E D hD
let K : ℝ := max 1 F
have hK : 1 ≤ K := le_max_left _ _
have hKF : (F:ℝ) ≤ K := le_max_right _ _
refine ⟨K,H,hK,hH,fun X hX S _ hnF _ I hI hMI hdim hdeg ha => ?_⟩
let : I.IsPrime := hI
by_cases hq : IsAlgebraic ℂ (Ideal.Quotient.mk I M.q)
· exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_polynomial_fibre_count S F M.q hnF I hq))
· exact Or.inr (hHilb I M.a M.b M.q M.θ M.distinct ha (M.quotient_equations I hMI)
M.generates hq hdim hdeg M.leftDegree M.qDegree w hw M.leftWeight M.rightWeight ω hω hscale)
· have he : h = d+1 := by omega
subst h
obtain ⟨H,hH,hHilb⟩ := norm_full_threshold_uniform (n := n) hd hd'
let K : ℝ := max 1 F
have hK : 1 ≤ K := le_max_left _ _
have hKF : (F:ℝ) ≤ K := le_max_right _ _
refine ⟨K,H,hK,hH,fun X hX S _ hnF _ I hI hMI hdim _ ha => ?_⟩
let : I.IsPrime := hI
by_cases hq : IsAlgebraic ℂ (Ideal.Quotient.mk I M.q)
· exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_polynomial_fibre_count S F M.q hnF I hq))
· right
simpa only [Nat.add_sub_cancel] using hHilb I M.a M.b M.q M.θ ha (M.quotient_equations I hMI)
M.generates hq hdim w hw M.leftWeight M.rightWeight ω hω hscale
end QuarticPowerFree
end OAI
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